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					NOVA: Einstein’s Big Idea – Special and General Relativity

In the middle of the 17th century, Sir Isaac Newton unified the Celestial and the Terrestrial with his
description of gravity as an influence between two objects, which can best be considered as a force
of attraction between objects as a result of the product of their masses and the square of the
distance between them. The effect of this influence is to cause the velocity of objects to change, in
accordance with Newton’s Laws of Motion. While his work in this area alone would have been
sufficient secure his place as one of the greatest scientists in history, he is also credited with a
number of achievements (see
     Inventing the reflecting telescope
     Proposed a new theory of light and color
     Discovered calculus
     Developed three laws of motion
     Devised the law of universal gravitation
     Advanced early modern chemisty
     Became the father of modern science

Independent of Newton’s work, Michael Faraday and James Clerk Maxwell collaborated to unify
electricity and magnetism, into the electro-magnetic force. In my interpretation of the historical
accounts, Michael Faraday was responsible for the majority of the science while James Clerk
Maxwell was responsible for the mathematics. One of the most important results of their efforts
was that electromagnetic energy can be thought of as a wave that travels at a single speed,
regardless of its frequency … the speed of light (see )

In his efforts to unify gravity and electro-magnetism, Albert Einstein made some of the most
incredible scientific advances in history. Through a variety of *thought experiments*, he
developed the theories of special and general relativity, each of which have been supported by a
variety of observations.

Complete the following after reading the attached materials and conducting additional research:
   Special Relativity:
          o Write a paragraph that explains Special Relativity in your words
          o State the Two Postulates of Special Relativity
          o Submit and explain an illustration that you have found helpful in improving your
              understanding of time dilation, length contraction, and/or the twin paradox (include
              the source)
   General Relativity:
          o Write a paragraph that explains General Relativity in your own words
          o State the Equivalence Principle
          o Submit and explain an illustration that you have found helpful in improving your
              understanding of gravitational bending of light, and/or gravitational time dilation
              (include the source)

The Theory Behind the Equation
                  by Michio Kaku
Imagine a police officer chasing after a speeding motorist. If he drives fast enough, the officer
knows that he can catch the motorist. Anyone who has ever gotten a ticket for speeding knows
that. But if we now replace the speeding motorist with a light beam, and an observer witnesses
the whole thing, then the observer concludes that the officer is speeding just behind the light
beam, traveling almost as fast as light. We are confident that the officer knows he is traveling
neck and neck with the light beam.

But later, when we interview him, we hear a strange tale. He claims that instead of riding
alongside the light beam as we just witnessed, it sped away from him, leaving him in the dust.
He says that no matter how much he gunned his engines, the light beam sped away at precisely
the same velocity. In fact, he swears that he could not even make a dent in catching up to the
light beam. No matter how fast he traveled, the light beam traveled away from him at the speed
of light, as if he were stationary instead of speeding in a police car.

But when you insist that you saw the police officer speeding neck and neck with the light beam,     Einstein realized that the world
within a hairsbreadth of catching up to it, he says you are crazy; he never even got close. To      described by Isaac Newton
Einstein, this was the central, nagging mystery: How was it possible for two people to see the      (left), in which one could add
same event in such totally different ways? If the speed of light was really a constant of nature,   and subtract velocities, and that
then how could a witness claim that the officer was neck and neck with the light beam, yet the      described by James Clerk
officer swears that he never even got close?                                                        Maxwell, in which the speed of
                                                                                                    light is constant, could not both
                                                                                                    be right. He decided to solve the
Einstein had realized earlier that the Newtonian picture (where velocities can be added and         problem—and special relativity
subtracted) and the Maxwellian picture (where the speed of light was constant) were in total        was the result.
contradiction. Newtonian theory was a self-contained system, resting on a few assumptions. If
only one of these assumptions were changed, it would unravel the entire theory in the same
way that a loose thread can unravel a sweater. That thread would be Einstein's daydream of
racing a light beam.


One day around May of 1905, Einstein went to visit his good friend Michele Besso, who also
worked at the patent office, and laid out the dimensions of the problem that had puzzled him for
a decade. Using Besso as his favorite sounding board for ideas, Einstein presented the issue:
Newtonian mechanics and Maxwell's equations, the two pillars of physics, were incompatible.
One or the other was wrong. Whichever theory proved to be correct, the final resolution would
require a vast reorganization of all of physics. He went over and over the paradox of racing a
light beam. Einstein would later recall, "The germ of the special relativity theory was already
present in that paradox." They talked for hours, discussing every aspect of the problem,
including Newton's concept of absolute space and time, which seemed to violate Maxwell's
constancy of the speed of light. Eventually, totally exhausted, Einstein announced that he was
defeated and would give up the entire quest. It was no use; he had failed.

Although Einstein was depressed, his thoughts were still churning in his mind when he returned
home that night. In particular, he remembered riding in a streetcar in Bern and looking back at
the famous clock tower that dominated the city. He then imagined what would happen if his
streetcar raced away from the clock tower at the speed of light. He quickly realized that the
clock would appear stopped, since light could not catch up to the streetcar, but his own clock in
the streetcar would beat normally.

Then it suddenly hit him, the key to the entire problem. Einstein recalled, "A storm broke loose
in my mind." The answer was simple and elegant: time can beat at different rates throughout
the universe, depending on how fast you moved. Imagine clocks scattered at different points in          Einstein in the Bern patent office
space, each one announcing a different time, each one ticking at a different rate. One second on        in 1904, just months away from
Earth was not the same length as one second on the moon or one second on Jupiter. In fact, the          the brilliant insight that led to
faster you moved, the more time slowed down. (Einstein once joked that in relativity theory, he         his theory of special relativity—
placed a clock at every point in the universe, each one running at a different rate, but in real life   and, a few weeks later, to E =
he didn't have enough money to buy even one.) This meant that events that were simultaneous             mc2
in one frame were not necessarily simultaneous in another frame, as Newton thought. He had
finally tapped into "God's thoughts." He would recall excitedly, "The solution came to me
suddenly with the thought that our concepts and laws of space and time can only claim validity
insofar as they stand in a clear relation to our experiences.... By a revision of the concept of
simultaneity into a more malleable form, I thus arrived at the theory of

“Thank you, I’ve completely solved the problem.”

For example, remember that in the paradox of the speeding motorist, the police officer was
traveling neck and neck with the speeding light beam, while the officer himself claimed that the
light beam was speeding away from him at precisely the speed of light, no matter how much he
gunned his engines. The only way to reconcile these two pictures is to have the brain of the
officer slow down. Time slows down for the policeman. If we could have seen the officer's
wristwatch from the roadside, we would have seen that it nearly stopped and that his facial
expressions were frozen in time. Thus, from our point of view, we saw him speeding neck and
neck with the light beam, but his clocks (and his brain) were nearly stopped. When we
interviewed the officer later, we found that he perceived the light beam to be speeding away,
only because his brain and clocks were running much slower.


The day after this revelation, Einstein went back to Besso's home and, without even saying
hello, he blurted out, "Thank you, I've completely solved the problem." He would proudly recall,
"An analysis of the concept of time was my solution. Time cannot be absolutely defined, and
there is an inseparable relation between time and signal velocity." For the next six weeks, he
                                                                                                        A streetcar trundles below the
furiously worked out every mathematical detail of his brilliant insight, leading to a paper that is
                                                                                                        clock tower in Bern that Einstein
arguably one of the most important scientific papers of all time. According to his son, he then
                                                                                                        made famous with his thought
went straight to bed for two weeks after giving the paper to his wife Mileva to check for any
                                                                                                        experiment about racing a light
mathematical errors. The final paper, "On the Electrodynamics of Moving Bodies," was scribbled
on 31 handwritten pages, but it changed world history.

In the paper, he does not acknowledge any other physicist; he only gives thanks to Michele
Besso. It was finally published in Annalen der Physik in September 1905, in volume 17. In fact,
Einstein would publish three of his pathbreaking papers in that famous volume 17. His colleague
Max Born has written, volume 17 is "one of the most remarkable volumes in the whole scientific
literature. It contains three papers by Einstein, each dealing with a different subject and each
today acknowledged to be a masterpiece." (Copies of that famous volume sold for $15,000 at
an auction in 1994.)

With almost breathtaking sweep, Einstein began his paper by proclaiming that his theories
worked not just for light, but were truths about the universe itself. Remarkably, he derived all
his work from two simple postulates applying to inertial frames (i.e., objects that move with
constant velocity with respect to each other):

    1.   The laws of physics are the same in all inertial frames.
    2.   The speed of light is a constant in all inertial frames.

These two deceptively simple principles mark the most profound insights into the nature of the
universe since Newton's work. From them, one can derive an entirely new picture of space and


First, in one masterful stroke, Einstein elegantly proved that if the speed of light was indeed a
constant of nature, then the most general solution was the Lorentz transformation*. He then
showed that Maxwell's equations did indeed respect that principle. Last, he showed that
velocities add in a peculiar way. Although Newton, observing the motion of sailing ships,
concluded that velocities could add without limit, Einstein concluded that the speed of light was
the ultimate velocity in the universe. Imagine, for a moment, that you are in a rocket speeding
at 90 percent the speed of light away from Earth. Now fire a bullet inside the rocket that is also
going at 90 percent the speed of light. According to Newtonian physics, the bullet should be
going at 180 percent the speed of light, thus exceeding light velocity. But Einstein showed that
because meter sticks are shortening and time is slowing down, the sum of these velocities is
actually close to 99 percent the speed of light. In fact, Einstein could show that no matter how
hard you tried, you could never boost yourself beyond the speed of light. Light velocity was the
ultimate speed limit in the universe.

We never see these bizarre distortions in our experience because we never travel near the
speed of light. For everyday velocities, Newton's laws are perfectly fine. This is the fundamental
reason why it took over 200 years to discover the first correction to Newton's laws. But now
imagine that the speed of light is only 20 miles per hour. If a car were to go down the street, it   Volume 17 of the German
might look compressed in the direction of motion, being squeezed like an accordion down to           physics journal Annalen der
perhaps one inch in length, for example, although its height would remain the same. Because          Physik, in which Einstein
the passengers in the car are compressed down to one inch, we might expect them to yell and          published no fewer than three
scream as their bones are crushed. In fact, the passengers see nothing wrong, since everything       groundbreaking papers at age
inside the car, including the atoms in their bodies, is squeezed as well.                            26.

As the car slows down to a stop, it would slowly expand from one inch to about 10 feet, and the
passengers would walk out as if nothing happened. Who is really compressed? You or the car?
According to relativity, you cannot tell, since the concept of length has no absolute meaning.


Einstein then pushed further and made the next fateful leap. He wrote a small paper, almost a
footnote, late in 1905 that would change world history. If meter sticks and clocks became
distorted the faster you moved, then everything you can measure with meter sticks and clocks
must also change, including matter and energy. In fact, matter and energy could change into
each other. For example, Einstein could show that the mass of an object increased the faster it
moved. (Its mass would in fact become infinite if you hit the speed of light—which is impossible,
which proves the unattainability of the speed of light.) This meant that the energy of motion
was somehow being transformed into increasing the mass of the object. Thus, matter and
energy are interchangeable. If you calculated precisely how much energy was being converted
into mass, in a few simple lines you could show that E = mc2, the most celebrated equation of
all time. Since the speed of light was a fantastically large number, and its square was even
larger, this meant that even a tiny amount of matter could release a fabulous amount of energy.
A few teaspoons of matter, for example, has the energy of several hydrogen bombs. In fact, a
piece of matter the size of a house might be enough to crack the Earth in half.

     “Imagine the audacity of such a step ... every speck of dust becoming a prodigious reservoir of
     untapped energy.”
                                                                                                                Other scientists came close to
                                                                                                                stumbling upon relativity before
                                                                                                                Einstein, including the Dutch
                                                                                                                physicist Hendrik Lorentz
Einstein's formula was not simply an academic exercise, because he believed that it might                       (seated fourth from left) and the
explain the curious fact discovered by Marie Curie, that just an ounce of radium emitted 4,000                  French mathematician Henri
calories of heat per hour indefinitely, seemingly violating the first law of thermodynamics (which              Poincaré (seated far right, next
states that the total amount of energy is always constant or conserved). He concluded that                      to Marie Curie). Einstein is
there should be a slight decrease in its mass as radium radiated away energy (an amount too                     standing second from right in
small to be measured using the equipment of 1905). "The idea is amusing and enticing; but                       this photo from a 1911
whether the Almighty is laughing at it and is leading me up the garden path—that I cannot                       conference.
know," he wrote. He concluded that a direct verification of his conjecture "for the time being
probably lies beyond the realm of possible experience."

Why hadn't this untapped energy been noticed before? He compared this to a fabulously rich
man who kept his wealth secret by never spending a cent.

Banesh Hoffman, a former student, wrote, "Imagine the audacity of such a step.... Every clod of
earth, every feather, every speck of dust becoming a prodigious reservoir of untapped energy.
There was no way of verifying this at the time. Yet in presenting his equation in 1907 Einstein
spoke of it as the most important consequence of his theory of relativity. His extraordinary
ability to see far ahead is shown by the fact that his equation was not verified ... until some
twenty-five years later."

Once again, the relativity principle forced a major revision in classical physics. Before, physicists
believed in the conservation of energy, the first law of thermodynamics, which states that the
total amount of energy can never be created or destroyed. Now physicists considered the total
combined amount of matter and energy as being conserved.

                                                                                                                The world's most famous
*Named for the Dutch physicist Hendrik Lorentz, who calculated them, the Lorentz                                equation, as it appears in
transformations are the distortions of space and time inherent in the equations for light, i.e.,                modified form in a manuscript
Maxwell's equations. These transformations state that the faster you move, the slower time                      on special relativity theory that
beats for you and the more compressed you become. (At the speed of light, hypothetically time                   Einstein wrote in 1912
would stop and distances would shrink to nothing, both of which are impossible.) These
transformations are necessary to keep the speed of light a constant in all inertial frames.

                    Michio Kaku, a theoretical physicist at the City University of New York, is the author of
Einstein's Cosmos: How Albert Einstein's Vision Transformed Our Understanding of Space and Time (Norton,
2004), from which this article was adapted with kind permission of the author and publisher

Relativity and the Cosmos
                 by Alan Lightman
In November of 1919, at the age of 40, Albert Einstein became an overnight celebrity,
thanks to a solar eclipse. An experiment had confirmed that light rays from distant stars
were deflected by the gravity of the sun in just the amount he had predicted in his theory
of gravity, general relativity. General relativity was the first major new theory of gravity
since Isaac Newton's more than 250 years earlier.

Einstein became a hero, and the myth-building began. Headlines appeared in newspapers
all over the world. On November 8, 1919, for example, the London Times had an article
headlined: "The Revolution In Science/Einstein Versus Newton." Two days later, The New
York Times' headlines read: "Lights All Askew In The Heavens/Men Of Science More Or
Less Agog Over Results Of Eclipse Observations/Einstein Theory Triumphs." The planet was
exhausted from World War I, eager for some sign of humankind's nobility, and suddenly
here was a modest scientific genius, seemingly interested only in pure intellectual pursuits.


What was general relativity? Einstein's earlier theory of time and space, special relativity,         If it were not for Einstein,
proposed that distance and time are not absolute. The ticking rate of a clock depends on              several decades might have
the motion of the observer of that clock; likewise for the length of a "yardstick." Published         passed before another physicist
in 1915, general relativity proposed that gravity, as well as motion, can affect the intervals        worked out the concepts and
of time and of space. The key idea of general relativity, called the equivalence principle, is        mathematics of general
that gravity pulling in one direction is completely equivalent to an acceleration in the              relativity, Lightman says.
opposite direction. A car accelerating forwards feels just like sideways gravity pushing you
back against your seat. An elevator accelerating upwards feels just like gravity pushing
you into the floor.

If gravity is equivalent to acceleration, and if motion affects measurements of time and
space (as shown in special relativity), then it follows that gravity does so as well. In
particular, the gravity of any mass, such as our sun, has the effect of warping the space
and time around it. For example, the angles of a triangle no longer add up to 180 degrees,
and clocks tick more slowly the closer they are to a gravitational mass like the sun.

Many of the predictions of general relativity, such as the bending of starlight by gravity
and a tiny shift in the orbit of the planet Mercury, have been quantitatively confirmed by
experiment. Two of the strangest predictions, impossible ever to completely confirm, are
the existence of black holes and the effect of gravity on the universe as a whole
COLLAPSED STARS                                                                                       einstein/rela-i.html to see:

A black hole is a region of space whose attractive gravitational force is so intense that no
                                                                                                   Einstein racing a light beam, a
matter, light, or communication of any kind can escape. A black hole would thus appear
                                                                                                      thought experiment that led him
black from the outside. (However, gas around a black hole can be very bright.) It is
                                                                                                      to special relativity;
believed that black holes form from the collapse of stars. As long as they are emitting heat
and light into space, stars are able to support themselves against their own inward gravity          Einstein in an elevator, which
with the outward pressure generated by heat from nuclear reactions in their deep interiors.           shows how gravity and
                                                                                                      acceleration are the same;
                                                                                                     and the sun warping space-time,
Every star, however, must eventually exhaust its nuclear fuel. When it does so, its
                                                                                                      a visualization of general
unbalanced self-gravitational attraction causes it to collapse. According to theory, if a
burned-out star has a mass larger than about three times the mass of our sun, no amount
of additional pressure can stave off total gravitational collapse. The star collapses to form a
black hole. For a nonrotating collapsed star, the size of the resulting black hole is
proportional to the mass of the parent star; a black hole with a mass three times that of
our sun would have a diameter of about 10 miles.

     General relativity may be the biggest leap of the scientific imagination in history.

The possibility that stars could collapse to form black holes was first theoretically
"discovered" in 1939 by J. Robert Oppenheimer and Hartland Snyder, who were
manipulating the equations of Einstein's general relativity. The first black hole believed to
be discovered in the physical world, as opposed to the mathematical world of pencil and
paper, was Cygnus X-1, about 7,000 light-years from Earth. (A light-year, the distance
light travels in a year, is about six trillion miles.) Cygnus X-1 was found in 1970. Since
then, a dozen excellent black hole candidates have been identified. Many astronomers and
astrophysicists believe that massive black holes, with sizes up to 10 million times that of
our sun, inhabit the centers of energetic galaxies and quasars and are responsible for their
enormous energy release. Ironically, Einstein himself did not believe in the existence of
black holes, even though they were predicted by his theory.                                     A swirling gas disk around a
                                                                                                probable black hole in M87
THE START OF EVERYTHING                                                                         Galaxy

Beginning in 1917, Einstein and others applied general relativity to the structure and
evolution of the universe as a whole. The leading cosmological theory, called the big bang
theory, was formulated in 1922 by the Russian mathematician and meteorologist
Alexander Friedmann. Friedmann began with Einstein's equations of general relativity and
found a solution to those equations in which the universe began in a state of extremely
high density and temperature (the so-called big bang) and then expanded in time, thinning
out and cooling as it did so. One of the most stunning successes of the big bang theory is
the prediction that the universe is approximately 10 billion years old, a result obtained
from the rate at which distant galaxies are flying away from each other. This prediction
accords with the age of the universe as obtained from very local methods, such as the
dating of radioactive rocks on Earth.

According to the big bang theory, the universe may keep expanding forever, if its inward
gravity is not sufficiently strong to counterbalance the outward motion of galaxies, or it
may reach a maximum point of expansion and then start collapsing, growing denser and
denser, gradually disrupting galaxies, stars, planets, people, and eventually even
individual atoms. Which of these two fates awaits our universe can be determined by
measuring the density of matter versus the rate of expansion. Much of modern cosmology,
including the construction of giant new telescopes such as the new Keck telescope in
Hawaii, has been an attempt to measure these two numbers with better and better
accuracy. With the present accuracy of measurement, the numbers suggest that our                An image of distant galaxies
universe will keep expanding forever, growing colder and colder, thinner and thinner.           taken by Hubble Deep Field

General relativity may be the biggest leap of the scientific imagination in history. Unlike
many previous scientific breakthroughs, such as the principle of natural selection, or the
discovery of the physical existence of atoms, general relativity had little foundation upon
the theories or experiments of the time. No one except Einstein was thinking of gravity as
equivalent to acceleration, as a geometrical phenomenon, as a bending of time and space.
Although it is impossible to know, many physicists believe that without Einstein, it could
have been another few decades or more before another physicist worked out the concepts
and mathematics of general relativity.

Note: This feature originally appeared on NOVA's "Einstein Revealed" Web site, which has
been subsumed into the "Einstein's Big Idea" Web site. Alan Lightman, a physicist and
novelist, is currently Adjunct Professor of Humanities at MIT. Some of his recent books are
Einstein's Dreams, The Diagnosis, Reunion, A Sense of the Mysterious, and The


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